Penrose type inequalities for asymptotically hyperbolic graphs

TL;DR

This paper establishes Penrose-type inequalities for asymptotically hyperbolic graphs, relating mass and boundary area under convexity conditions, inspired by Euclidean analogs and extending geometric inequalities in hyperbolic settings.

Contribution

It introduces new Penrose-type inequalities for asymptotically hyperbolic graphs, linking mass estimates to boundary geometry under convexity assumptions.

Findings

Mass can be estimated via boundary integrals involving scalar curvature.

Convexity of the inner boundary allows area-based bounds.

Results extend Penrose inequalities to hyperbolic manifolds.

Abstract

In this paper we study asymptotically hyperbolic manifolds given as graphs of asymptotically constant functions over hyperbolic space $\bH^n$. The graphs are considered as subsets of $\bH^{n+1}$ and carry the induced metric. For such manifolds the scalar curvature appears in the divergence of a 1-form involving the integrand for the asymptotically hyperbolic mass. Integrating this divergence we estimate the mass by an integral over an inner boundary. In case the inner boundary satisfies a convexity condition this can in turn be estimated in terms of the area of the inner boundary. The resulting estimates are similar to the conjectured Penrose inequality for asymptotically hyperbolic manifolds. The work presented here is inspired by Lam's article concerning the asymptotically Euclidean case.